1. Introduction
Due to the constant renovation of the trend of real-world environmental change, distribution mechanisms of a combination of different theoretical fields, including sports usability distribution, have become a major conception in the context of sustainability. In the framework of sports usability distribution, the use of several notions could promote the distribution efficiency no matter for improvement of distribution methods or sports management techniques of usability. On the other hand, the axiomatic outcomes of game-theoretical distribution mechanisms could be always adopted to analyze various interaction relationships and related models among agents and coalitions by applying mathematical outcomes. In addition to theoretical analysis, game-theoretical outcomes also have been applied to offer optimal outcomes or equilibrium situations for many real-word models. Based on the demands for accounting, economics, management sciences and even sports sciences, some more power outputs are introduced as adequate mappings from weights and decision quota to power. The most extensively applied one is the Banzhaf–Coleman output. The Banzhaf–Coleman output, named after Banzhaf [
1], is a power output proposed by the probability of changing a result of a vote where voting interests are not necessarily distributed equally among the voters. Plainly speaking, the Banzhaf–Coleman output is a distribution notion that collects each performer’s average marginal contribution from all participated coalitions. Several related outcomes may be found in, e.g., Banzhaf [
1], Owen [
2], Dubey and Shapley [
3], Moulin [
4], Lehrer [
5], Haller [
6] and van den Brink and van der Laan [
7].
Under traditional schemes, each performer is either entirely participated or not participated at all in participation with other performers, while under a multi-choice scheme, each performer could be consented to take finite many different energy levels. It is known that outputs on multi-choice schemes could be performed under various issues such as economics, management sciences, sports sciences and so on. Hwang and Liao [
8,
9] and Liao [
10] proposed several extensions of the max-reduced scheme and the complement-reduced scheme to characterize several core concepts in the context of multi-choice schemes. Later, Hwang and Liao [
11] also considered an extended self-reduced scheme to analyze a multi-choice generalization of the Shapley value.
In different topics, from biomedical engineering, sciences to environment and the management sciences, performers confront an increasing demand to focus on multiple objectives effectively in its operational procedures. Related conditions include analyzing allocation tradeoffs, selecting optimal strategy or course designs or an arbitrary other situation where you need an effective rule with tradeoffs among several objectives. Under various cases, these real-world effective conditions might be modeled as a mathematical multi-attribute optimization status. The rules of such conditions require appropriate techniques to present optimal outcomes that—unlike traditional notions or viewpoints—apply several properties of the objectives into account.
Under the axiomatic processes for outputs under cooperative schemes,
consonance is a crucial property of useful outputs. The notion behind this type of consonance is as follows: for a given scheme, performers might develop prospects of the scheme and may be willing to consent the computation of its remunerations to be based upon these prospects. The output concept is consonant if it gives the same remunerations to performers in the original scheme as it does to performers of the imaginary reduced scheme. Thus, consonance is a requisite of the inner “robustness" of compromises. In addition to axiomatic characterizations,
dynamic processes can be represented that lead the performers to that output, commencing from an arbitrary efficacious remuneration vector. The main base of a dynamic notion was raised from Stearns [
12].
The above-mentioned existing-outcomes generate one motivation:
Different from the contexts of traditional schemes and multi-choice schemes, we focus on the framework of multi-attribute multi-choice schemes throughout this paper.
In
Section 2.1, the
multi-consideration Banzhaf–Coleman output, the
multi-consideration efficient Banzhaf–Coleman output and the
multi-consideration normalized Banzhaf–Coleman output are further defined by adopting maximal-usability among multi-choice level vectors on multi-attribute multi-choice schemes.
In
Section 2.1, some motivating and practical examples are provided to present related applications for sports management.
By applying an extended reduction, we propose some axiomatic processes to present the reasonability for these outputs in
Section 3. In order to analyze the dynamic processes of these outputs, we adopt alternative formulations for these outputs in terms of
excess mappings.
In
Section 4, specific reductions and excess mapping are applied to present that these outputs can be reached by performers who commence from an arbitrary efficacious remuneration vector.
2. Preliminaries
2.1. Definitions and Notations
Let be the universe of performers and be a set of performers. Let with be the vector that shows the amount of energy levels for each performer, at which it can operate. For , we define to be the level repository of performer p, where 0 means not acting, and . For , , let be the product set of the level repositories for performers in P. For every , we define as the vector with if , and if . Denote to be the zero vector in . For , let .
A multi-choice scheme is denoted by , where is a finite collection of performers, is the vector that appears the amount of energy levels for each performer and is a map which apportions to each the usability that the performers can accept when each performer p takes energy level with . A scheme will be denoted by the map h if there is no confusion. Given a multi-choice scheme and , we denote to be the multi-choice subscheme defined by restricting h to . A multi-attribute multi-choice scheme is a triple , where , and is a multi-choice scheme for every . Let be the collection of all multi-attribute multi-choice schemes.
Given and , we define that and to be the restriction of at K for each . Further, we define to be the maximal-usability (Here we apply bounded multi-choice schemes, defined as such that, there exists such that for every . We apply it to guarantee that is well-defined.) among all vectors with . A remuneration vector of is a vector and , where denotes the remuneration to performer p in for every and for every . A remuneration vector x of is multi-attribute efficacious if for every . The collection of all multi-attribute efficacious vectors of is denoted by .
An output is a map
apportioning to each
, an element
where
and
is the remuneration of the performer
p apportioned by
in
.
Next, we introduce three outputs under the multi-attribute multi-choice situation.
Definition 1. The multi-consideration Banzhaf–Coleman output (MBCO), Θ
, is defined to be for every , for every and for every , Under the output Θ, all performers receive an average marginal contribution of maximal-usability in each .
An output conforms multi-attribute efficacy (MEIY) if for every and for every , . MEIY presents that all performers completely allocate all the usability. Clearly, the MBCO violates MEIY. In the following, we consider an efficacious extension and an normalization.
Definition 2. The multi-consideration efficacious Banzhaf–Coleman output (MEBCO), , is defined for every , for every and for every , Under the output , all performers first receive its MBCO, and further allocate the remaining usability equally.
The multi-consideration normalized Banzhaf–Coleman output (MNBCO), , is defined for every , for every and for every , where . Under the notion of , all performers allocate the maximal-usability of the grand coalition proportionally by applying the MBCO of all performers.
Lemma 1. The MEBCO and the MNBCO conform MEIY on Δ and , respectively.
Proof. For every
and for every
,
Thus, the MEBCO conforms MEIY on
. For every
and for every
,
Thus, the MNBCO conforms MEIY on . □
2.2. Motivating and Practical Examples
As we mentioned in the introduction, each performer might be allowed to participate with different levels in real situations respectively. On the other hand, multi-attribute analysis is a notion of multiple criterion analysis that is concerned with situations involving simultaneously more than one objective to be optimized. Multi-attribute analysis has been adopted in many issues, including biomedical engineering, economics, politics, sports management sciences, logistics, where efficacious decisions need to be adopted in the presence of trade-offs among several objectives. For instance, minimizing cost while maximizing comfort while marketing a central air conditioning system, and maximizing efficacy whilst minimizing energy consumption and emission of pollutants are examples of multi-attribute efficacious problems involving respectively two and three objectives. Under various cases, there might be more than three objectives. Hence, we focus on the framework of multi-attribute multi-choice schemes throughout this paper. The advantages of our methods are that these outputs of the multi-attribute multi-choice scheme always exist and result in a kind of global outcome for a specific performer by summarizing the result under all its energy levels.
Here we provide a brief motivating example of multi-choice schemes in the setting of “sports management”. Let be a set of all performers of a sports management system . The function h could be treated as a usability function which assigns to each level vector the worth that the performers can obtain when each performer p participates at operation strategy in . Modeled in this way, the sports management system could be considered as a multi-choice scheme, with h being each characteristic function and being the set of all operation strategies of the performer p.
In the following, we also provide a practical application of power distribution in a sports association, such as NBA, MLB and so on. Let be a set of all performers of a sports association. In the sports association, all the performers are elected by voting or recommendation by sports parties (or teams). All performers have the power to propose, discuss, establish and veto all bills (or rules). They dedicate different levels of attention and participation to different bills depending on their academic expertise and the public opinion they represent. The level of involvement is also closely associated with the alliance strategy formed for the interests of different sports parties. For the aforementioned reasons, strategies adopted by each performer of the parliament show distinct levels of participation and certain amounts of ambiguity. The function h could be treated as a power function which assigns to each level vector the power that the performers can dedicate when each performer p participates at operation strategy . Modeled in this way, the sports association operational system could be considered as a multi-choice scheme, with h being each characteristic function and being the set of all operation strategies of the performer p. To evaluate the influence of each performer in the sports association, using the power indicators we proposed, we first assess the influence each sports association performer has arisen over previous bill meetings based on various strategies, which are the outputs mentioned in Definitions 1 and 2.
Here we provide an application with real data as follows. Let
with
,
and
. Thus,
. Further, let
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
and
. Therefore, we have that
,
,
,
,
,
,
,
,
,
,
,
,
,
and
. By Definitions 1 and 2,
In the following sections, we would like to demonstrate that the MBCO, the MEBCO and the MNBCO could present “optimal distribution mechanisms” among all performers, in the sense that this organization can receive remuneration from each combination of operational levels of all performers under multi-choice behavior and multi-attribute situations.
3. Axiomatic Outcomes
Here, we demonstrate that there exists corresponding reduced schemes that could be applied to axiomatize the MBCO, the MEBCO and the MNBCO.
Let
be an output,
and
. The 1-reduced scheme
is defined by
and
for every
. An output
conforms 1-consonance (1CSE) if
for every
, for every
with
, for every
and for every
. Further,
conforms 1-criterion for schemes (1CS) if
for every
with
. In the following, we characterize the MBCO by adopting 1CSE and 1CS.
Lemma 2. - 1.
The MBCO conforms 1CSE on Δ.
- 2.
On Δ, the MBCO is the only output conforming 1CS and 1CSE.
Proof. To demonstrate result 1, let
and
. The proof is finished if
. Suppose that
and
for some
, for every
and for every
,
Hence, conforms 1CSE.
By result 1,
conforms 1CSE on
. Clearly,
conforms 1CS on
. To demonstrate uniqueness of result 2, suppose that
conforms 1CS and 1CSE. Let
. If
, then
by 1CS. The condition
: Let
and
, and let
with
and
. Therefore,
Thus, . □
In the following we take some examples to manifest that each of the properties applied in Lemma 2 are independent of the rest of properties.
Example 1. Consider an output ρ by for every , for every and for every , . Clearly, ρ conforms 1CSE, but it does not conform 1CS.
Example 2. Consider an output ρ for every , for every and for every , On Δ, ρ conforms 1CS, but it does not conform 1CSE.
It is easy to demonstrate that the output
and
violates 1CSE. In the following, we consider the
2-reduced scheme. Let
be an output,
and
. The 2-reduced scheme
is defined by
and
for every
. An output
conforms 2-consonance (2CSE) if
for every
, for every
with
, for every
and for every
. Furthermore,
conforms 2-criterion for schemes (2CS) if
for every
with
.
Clearly, does not exist if . Thus, we consider the 3-consonance as follows. An output conforms 3-consonance (3CSE) if for some and for some with , it holds that for every and for every . Further, conforms 3-criterion for schemes (3CS) if for every with .
Subsequently, we characterize the MEBCO and the MNBCO by respectively adopting 2CSE, 3CSE, 2CS and 3CS. In order to establish consonance of the MEBCO and the MNBCO, it will be useful to introduce alternative formulation for the MEBCO and the MNBCO in terms of
excess. Let
,
and
x be a remuneration vector in
. Define that
for every
. The excess of a coalition
at
x is considered to be
The value can be regarded as the complaint of coalition S when all performers in S receive its remunerations from in .
Lemma 3. For every , for every , for every and for every , Proof. Let
and
. For every
and for every
,
By Equations (
2) and (
3), for every
,
That is, . Since and conforms MEIY, . Hence, for every and for every , i.e., . □
Remark 1. for every and for every .
Theorem 1. - 1.
The MEBCO conforms 2CSE on Δ.
- 2.
If ρ conforms 2CS and 2CSE, then it also conforms MEIY.
- 3.
On Δ, the MEBCO is the only output conforming 2CS and 2CSE.
Proof. To demonstrate result 1, let
and
. The proof is finished if
. Suppose that
,
and
for some
. For every
and for every
,
Since
conforms MEIY,
by definition of 2-reduced scheme. Further, by Lemma 3 and Equation (
4),
Therefore, . That is, conforms 2CSE.
To demonstrate result 2, suppose
conforms 2CS and 2CSE. Let
and
. By 2CS,
conforms MEIY if
. The condition
: Suppose, on the contrary, that there exists
such that
. This presents that there exists
and
such that
. By 2CSE,
conforms MEIY for two-person schemes and this contradicts with
Hence, conforms MEIY.
To demonstrate result 3,
conforms 2CSE by result 1. Clearly,
conforms 2CS. To demonstrate uniqueness, suppose
conforms 2CS and 2CSE; hence, by result 2,
also conforms MEIY. Let
. By 2CS,
if
. The condition
: Let
,
and
for some
. Then,
By definitions of
and
,
By Equation (
6), Equation (
5) becomes
That is, for every
,
Thus, for every , i.e., . □
In the following we take some examples to show that each of the properties applied in Theorem 1 are independent of the rest of properties.
Example 3. Consider an output ρ for every , for every and for every , . Clearly, ρ conforms 2CSE, but it does not conform 2CS.
Example 4. Consider an output ρ for every , for every and for every , On Δ, ρ conforms 2CS, but it does not conform 2CSE.
Lemma 4. Let and . Then, for every and , where .
Proof. Let
and
. For every
and
,
By Equations (
7) and (
8), for every pair
and
,
That is, . Since and conforms MEIY, . Hence, for every , for every and for every . □
Theorem 2. - 1.
The MNBCO conforms 3CSE on .
- 2.
If ρ conforms 3CS and 3CSE, then it also conforms MEIY.
- 3.
On , the MNBCO is the only output conforming 3CS and 3CSE.
Proof. The proofs of outcomes 1 and 2 are similar to the proofs of outcomes 1 and 2 of Theorem 1, so we omit them. Next, we demonstrate result 3. By result 1, the MNBCO conforms 3CSE. Clearly, the MNBCO conforms 3CS. To demonstrate uniqueness, suppose
conforms 3CSE and 3CS on
. By result 2,
conforms MEIY on
. Let
. The proof could be finished by induction on
. By 3CS, it is trivial that
if
. Suppose that it holds if
,
. The condition
: Let
with
and
. By Definition 2,
for every
. Assume that
for every
and for every
. Therefore,
The proof is finished. □
In the following we take some examples to manifest that each of the properties applied in Theorem 1 are independent of the rest of the properties.
Example 5. Consider an output ρ by for every , for every and for every , . On , ρ conforms 3CSE, but it does not conform 3CS.
Example 6. Consider an output ρ by for every , for every and for every , On , ρ conforms 3CS, but it does not conform 3CSE.
4. Dynamic Processes
Here, we adopt excess mappings and reductions to offer dynamic outcomes for the MEBCO and the MNBCO.
In order to present the dynamic processes of the MEBCO and the MNBCO, we firstly define switch mappings by means of excess mappings. The switch mappings are based on the notion that each performer shortens the complaint relating to its own and others’ non-participation, and uses these regulations to revise the original remuneration. In the following, we firstly introduce the dynamic processes for the MEBCO.
Definition 3. Let and . Theswitch mappingis considered to be , where and is defined by Define for every .
Lemma 5. for every and for every .
Proof. Let
,
,
and
. Similar to Equation (
2),
By Equations (
10) and (
11),
Hence, if . □
Theorem 3. Let . If , then converges to for each .
Proof. Let
,
,
and
. By Equation (
12) and the definition of
f,
Therefore, for every
,
If , then and converges to . □
Similar to the work of Maschler and Owen [
13], a dynamic outcome could be provided under reductions as follows.
Definition 4. Let ρ be an output, , and . The -reduced scheme is given by and for every , Similar to the work of Maschler and Owen [
13], we also introduce related switch mapping as follows. The R-switch mapping is
, where
and
is defined by
Define for every .
Lemma 6. for every and for every .
Proof. Let
,
,
and
. Let
, by MEIY of
and Definition 4,
By definition of
g and Equation (
14),
Thus, for every . □
Theorem 4. Let . If , then converges to for each .
Proof. Let
,
and
. By Equation (
15),
for every
. Therefore,
Therefore, for every
,
If , then and converges to for every , for every and for every . □
Subsequently, we present the dynamic outcomes for the MNBCO.
Definition 5. Let and . The N-switch mapping is considered to be , where and is considered by where . Let for every .
The N-R-switch mapping is , where and is considered by Let for every .
Theorem 5. and for every and for every .
Let . If , then converges to for each .
Let . If , then converges to for each .
Proof. The proofs of this theorem are similar to Lemmas 5, 6 and Theorems 3 and 4. □